Finite Group Databasebeta

← H(7,𝔽2)×C137 #31778605

Subgroups — H(7,𝔽2)×C137 #31778605

Order 17536.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C137C2128yesmaximal, complemented
2C2×H(5,𝔽2)×C137C2135yesmaximal, complemented
32−1+4×C137C22156yescomplemented
4H(5,𝔽2)×C137C221280yescomplemented
5C2×#41×C137C221210yescomplemented
6C22×D4×C137C221105yescomplemented
7C24×C137C23130yescomplemented
8#41×C137C231560yescomplemented
9C2×Q8×C137C23170yescomplemented
10C2×D4×C137C231630yescomplemented
11C22×C4×C137C231105yescomplemented
12C23×C137—830no—
13C23×C137C241105yes—
14Q8×C137C24156yes—
15D4×C137C241280yes—
16C2×C4×C137C241210yes—
17C22×C137—4105no—
18C22×C137C25135yes—
19C4×C137C25128yes—
20C2×C137—235no—
21C2×C137C2611yescharacteristic
22C137H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
23H(7,𝔽2)C13711yesmaximal, characteristic, direct product factor, Sylow
24#585C2×C137128yescomplemented
25C2×H(5,𝔽2)C2×C137135yescomplemented
262−1+4C22×C137156yescomplemented
27H(5,𝔽2)C22×C1371280yescomplemented
28C2×#41C22×C1371210yescomplemented
29C22×D4C22×C1371105yescomplemented
30C24C23×C137130yescomplemented
31#41C23×C1371560yescomplemented
32C2×Q8C23×C137170yescomplemented
33C2×D4C23×C1371630yescomplemented
34C22×C4C23×C1371105yescomplemented
35C23—830no—
36C23C24×C1371105yes—
37Q8C24×C137156yes—
38D4C24×C1371280yes—
39C2×C4C24×C1371210yes—
40C22—4105no—
41C22C25×C137135yes—
42C4C25×C137128yes—
43C2—235no—
44C2C26×C13711yescharacteristic
subgroupG/H
center, socleC2×C137C26
commutator, Frattini, cosocleC2C26×C137
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #585×C137C2semidirect
2. C2×H(5,𝔽2)×C137C2semidirect
3. 2−1+4×C137C22semidirect
4. H(5,𝔽2)×C137C22semidirect
5. C2×#41×C137C22semidirect
6. C22×D4×C137C22semidirect
7. C24×C137C23semidirect
8. #41×C137C23semidirect
9. C2×Q8×C137C23semidirect
10. C2×D4×C137C23semidirect
11. C22×C4×C137C23semidirect
22. C137H(7,𝔽2)direct (N × H)
23. H(7,𝔽2)C137direct (N × H)
24. #585C2×C137semidirect
25. C2×H(5,𝔽2)C2×C137semidirect
26. 2−1+4C22×C137semidirect
27. H(5,𝔽2)C22×C137semidirect
28. C2×#41C22×C137semidirect
29. C22×D4C22×C137semidirect
30. C24C23×C137semidirect
31. #41C23×C137semidirect
32. C2×Q8C23×C137semidirect
33. C2×D4C23×C137semidirect
34. C22×C4C23×C137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#585×C1372
C2×H(5,𝔽2)×C1372
H(7,𝔽2)137